Problem 45 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{2}{3 x}+\frac{4}{6 x^{2}} $$

Step-by-Step Solution

Answer

Question: Add the two given rational expressions: $$\frac{2}{3x} + \frac{4}{6x^2}$$ Answer: The sum of the two given rational expressions is: $$\frac{2}{3x} + \frac{4}{6x^2} = \frac{2(x+1)}{3x^2}$$

Step 1: Find the LCM

To find the LCM, we'll first look at the prime factors of each denominator. The first one is \(3x\) (with prime factors 3 and x) and the second one is \(6x^2\) (with prime factors 2, 3, and x^2). The LCM will include the highest powers of each prime factor found in the denominators. In this case, the LCM will be \(2\times3\times x^{2} = 6x^2\).

Step 2: Rewrite each fraction with the LCM

Now, we'll rewrite each fraction with the LCM as a denominator. We need to see what factor we have to multiply each denominator in order to make it equal to the LCM and then multiply it with the corresponding numerator. \\ For the first fraction: \\ $$ \frac{2}{3x} = \frac{2\times(2x)}{3x\times(2x)} = \frac{4x}{6x^2} $$ For the second fraction: \\ $$ \frac{4}{6x^2} = \frac{4}{6x^2} $$

Step 3: Add the numerators

Now that both fractions have the same denominator, we can add the numerators: \\ $$ \frac{4x}{6x^2} + \frac{4}{6x^2} = \frac{4x+4}{6x^2} $$

Step 4: Simplify the final fraction if possible

The last step is to simplify the final result. We can factor out a 4 from the numerator: \\ $$ \frac{4x+4}{6x^2} = \frac{4(x+1)}{6x^2} $$ Now, divide both numerator and denominator by their greatest common divisor (GCD), which is 2: \\ $$ \frac{4(x+1)}{6x^2} = \frac{2(x+1)}{3x^2} $$ The final answer is: $$ \frac{2}{3x} + \frac{4}{6x^2} = \frac{2(x+1)}{3x^2} $$

Key Concepts

Least Common Multiple (LCM)

When adding or subtracting fractions, it is crucial to have a common denominator. The Least Common Multiple (LCM) assists in finding that common denominator. Let's break it down.
First, identify the prime factors of each denominator. In our example:

• The first denominator is \(3x\), made of the prime factors 3 and \(x\).
• The second denominator is \(6x^2\), made of prime factors 2, 3, and \(x^2\).

To get the LCM, include the highest power of each prime factor present:

• The highest power of 2 is 2.
• For 3, it remains 3 because it appears once in both terms.
• For \(x\), it is \(x^2\) since that is the largest power seen.

Combine these to get the LCM: \(2 \times 3 \times x^2 = 6x^2\). This will be the new common denominator you need for the rational expressions.

Simplifying Fractions

Simplifying fractions is all about reducing them to their simplest form. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing through by this number.

Consider the fraction \(\frac{4(x+1)}{6x^2}\). The numbers 4 and 6 have a GCD of 2, which means both can be evenly divided by 2. Here's how you do it:

• Divide the numerator by 2 to get \(2(x+1)\).
• Divide the denominator by 2 to get \(3x^2\).

This results in \(\frac{2(x+1)}{3x^2}\), a more simplified form.
Simplifying makes fractions easier to handle and understand. Be sure to simplify whenever possible, especially as the final step!

Prime Factorization

Prime factorization is the process of breaking down a number or expression into the product of its basic building blocks, or 'prime' components. Each of these is a prime number or variable that can't be reduced further without losing its identity.
Let's use our example for clarification. The number 6 becomes 2 and 3, since \(6 = 2 \times 3\). Now incorporate variables:

• The term \(3x\) becomes 3 \(\times x\) because \(x\) is just a variable.
• The term \(6x^2\) becomes \(2 \times 3 \times x^2\).

Once you break down the terms into their prime factors, you can use this information to easily find the LCM.

Prime factorization is essential, especially with polynomials and variables. It gives you a fundamental understanding that aids in operations like determining least common multiples or even simplifying complex expressions.